Suppose that $F$ and $X$ are events from a common sample space with $P(F) \neq 0$ and $P(X) \neq 0$.
a.) Prove that $P(X) = P(X|F)P(F) + P(X|\bar{F})P(\bar{F})$. Hint: Explain why $P(X|F)P(F) = P(X \cap F)$ is another way of writing the definition of conditional probability, and then use that with the logic from the proof of Theorem
b.) Explain why $P(F|X) = P(X|F)P(F)/P(X)$ is another way of stating Bayes’ Theorem.
Having a bit of trouble with this question. Bayes Theorem hasn't really clicked with me yet. Can someone help me answer this question? And possibly help me better understand Bayes Theorem? I really appreciate all the help.
Edit: I really feel like the more I work on this problem the more I confuse myself I can prove the theory works when I use a made up example of events, but I just can't grasp what I need to do to answer this problem correctly
Intuitively, part (a.) is just the fact that $P(X)$ can be decomposed into two parts: Where $F$ happens and where $F$ doesn't happen. That is to say $P(X) = P(X\cap F \cup X\cap \overline F) = P(X\cap F)+P(X\cap \overline F)$. But then $P(X\cap F)=P(F)P(X|F)$ and likewise for $\overline F$.